Here are some comments to your code:

• You don't need to open the input file before passing it to $extor->parse_file
• Use the glob function (glob("./*.html)) instead of readdir and parse the resulting files for the ".html" extension
• You are not extracting the links (you are not using the method $extor->links)
• You are using $_ inside the foreach loop, but that variable is not initialized (because you are using a "loop variable" ($file))
• $file is declared outside the foreach loop, you should avoid this and restrict the scope of the variable at the minimum
• You should get the habit of closing the filehandles

Here is a version of your code with all these issues corrected (untested):

use strict;
use warnings;
use HTML::SimpleLinkExtor;

my $FileOut = "output.txt";

open FileDOut, ">$FileOut" or die "Can't open output file $FileOut: $!
+";
print "Summary written in file $FileOut\n";

while (my $file = glob ("*.html")){
  print FileDOut "$file\n";
  my $extor = HTML::SimpleLinkExtor->new();
  $extor->parse_file($file);
  my @links = $extor->links();
  print FileDOut "$_\n" for (@links);
  print FileDOut "\n";
}

close FileDOut;
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